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+%The following command typeset the series of the second
+%column of the page 9
+%
+%This command has one parameter:
+% 1) The width of the mathematical text
+\newcommand\TNineSeriesThree[1]{%
+ %This command typeset a fact about a serie and
+ %the math used to describe the property.
+ %
+ %The command has two parameters:
+ % 1) The title of the fact
+ % 2) The corresponding math.
+ %This macro uses the global (to the macro) variables '\Hspace', '\CurrentLineWidth'
+ %which should contain the space used to typeset the math.
+ \def\SerieProperty##1##2{%
+ \def\temp{\TNineTitle{##1} \ensuremath{##2}}%
+ \savebox\TmpBoxA{\temp}%
+ \settowidth\CurrentLineWidth{\usebox\TmpBoxA}%
+ \ifdimcomp{\CurrentLineWidth}{<}{\HSpace}%
+ {%Enough space, everything is written on one line
+ \temp
+ }{%The title is on one line, the math typeset as 'display math'
+ \TNineTitle{##1}%
+ \begin{displaymath}%
+ ##2%
+ \end{displaymath}%
+ }%
+ \par
+ \AdjustSpace{\TNineInterTitle}%
+ }%
+ %This command typeset a line showing the expansion of ordinary power serie
+ %
+ %The command has 2 parameters
+ % 1) The formula to be expanded
+ % 2) The expanded version
+ \def\LineOfOrdPowSerie##1##2{##1&=&##2\\[\TNineExpansionSkip]}%
+ \parbox[t]{#1}{%
+ \TNineSeriesFontSize
+ \DisplaySpace{\TNineDisplaySpace}{\TNineDisplayShortSpace}%
+ \setlength{\HSpace}{#1}%
+ \SerieProperty{Ordinary power series:}
+ {A(x) = \sum_{i=0}^\infty a_i x^i}
+ \SerieProperty{Exponential power series:}
+ {A(x) = \sum_{i=0}^\infty a_i \frac{x^i}{i!}}
+ \SerieProperty{Dirichlet power series:}
+ {A(x) = \sum_{i=1}^\infty \frac{a_i}{i^x}}
+ \SerieProperty{Binomial theorem:}
+ {(x + y)^n = \sum_{k=0}^n\binom{n}{k} x^{n-k} y^k}
+ \SerieProperty{Difference of like powers:}
+ {x^n - y^n = (x-y) \sum_{k=0}^{n-1} x^{n-1-k} y^k}
+ \SerieProperty{For ordinary power series:}
+ {\begin{array}{lcl}
+ \LineOfOrdPowSerie{\alpha A(x) + \beta B(x)}%
+ {\sum_{i=0}^\infty(\alpha a_i + \beta b_i) x^i}
+ \LineOfOrdPowSerie{x^k A(x)}%
+ {\sum_{i=k}^\infty a_{i-k} x^i}
+ \LineOfOrdPowSerie{\frac{A(x) - \sum_{i=0}^{k-1} a_i x^i}{x^k}}%
+ {\sum_{i=0}^\infty a_{i+k} x^i}
+ \LineOfOrdPowSerie{A(cx)}%
+ {\sum_{i=0}^\infty c^i a_i x^i}
+ \LineOfOrdPowSerie{A'(x)}%
+ {\sum_{i=0}^\infty (i+1) a_{i+1} x^i}
+ \LineOfOrdPowSerie{x A'(x)}%
+ {\sum_{i=1}^\infty i a_i x^i}
+ \LineOfOrdPowSerie{\int A(x) \, dx}%
+ {\sum_{i=1}^\infty \frac{a_{i-1}}{i} x^i}
+ \LineOfOrdPowSerie{\frac{A(x) + A(-x)}{2}}%
+ {\sum_{i=0}^\infty a_{2i} x^{2i}}
+ \LineOfOrdPowSerie{\frac{A(x) - A(-x)}{2}}%
+ {\sum_{i=0}^\infty a_{2i+1} x^{2i+1}}
+ \end{array}
+ }
+ \SerieProperty{Summation:}
+ {\text{If }b_i = \sum_{j=0}^i a_i\text{ then }
+ B(x) = \frac{1}{1 -x} A(x)}
+ \SerieProperty{Convolution:}
+ {A(x)B(x) = \sum_{i=0}^\infty\left(\sum_{j=0}^i a_j b_{i-j}\right)x^i}
+ }
+}